From point A to a circle with center O and a radius of 8 cm, draw tangents AB and AC, angle BAC = 60 degrees.

univerkov | January 4, 2021 | education

From point A to a circle with center O and a radius of 8 cm, draw tangents AB and AC, angle BAC = 60 degrees. Find AB and AC.

The radii OB and OС are perpendicular to the tangents AB and AC, then in the right-angled triangles AOB and AOC, OB = OС = R = 8 cm, the hypotenuse of OA is common, which means that the triangles AOB and AOC are equal along the leg and hypotenuse, and then the angle OAB = OAC = BAC / 2 = 60/2 = 300.
The OB leg lies opposite an angle of 300, then OA = 2 * OB = 2 * 8 = 16 cm.
By the Pythagorean theorem, AB2 = OA2 – OB2 = 256 – 64 = 192.
AB = 8 * √3 cm.
AC = AB = 8 * √3 cm.
Answer: The length of the segments AB and AC is 8 * √3 cm.

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